CONFORMAL MODULUS ON DOMAINS WITH STRONGSINGULARITIES AND CUSPS
HARRI HAKULA
∗
, ANTTI RASILA
†
,
AND
MATTI VUORINEN
‡
Abstract.
We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numericalsolution of the associated Dirichlet and DirichletNeumann type boundary values problems for theLaplace equation. Several experimental results, with error estimates, are reported. In particular,we consider domains with dendrite like boundaries, in such cases where an analytic formula for theconformal modulus can be derived. Our numerical method makes use of an
hp
FEM algorithm,written for this very complicated geometry with strong singularities.
Key words.
conformal capacity, conformal modulus, quadrilateral modulus,
hp
FEM, numericalconformal mapping
AMS subject classiﬁcations.
65E05, 31A15, 30C85
FILE: hrv3.tex printed: 2015128, 1.32
1. Introduction.
The conformal modulus is an important tool in geometricfunction theory [1], and it is closely related to certain physical quantities which alsooccur in engineering applications. For example, the conformal modulus plays animportant role in determining resistance values of integrated circuit networks (seee.g. [36, 40]). We consider both simply and doublyconnected bounded domains. By
deﬁnition such a domain can be mapped conformally either onto a rectangle or ontoan annulus, respectively. For the numerical study of these two cases we deﬁne themodulus
h
as follows. In the simply connected case, we ﬁx four points on the boundaryof the domain, call a domain with these ﬁxed boundary points a quadrilateral, andrequire that these four points are mapped onto vertices (0,0), (1,0), (1,h), (0,h) of the rectangle. In the doubly connected case we require that the annulus is
{
(
x,y
) :exp(
−
h
)
< x
2
+
y
2
<
1
}
.
Doubly connected domains are also called ring domains orsimply rings. Surveys of the state of the art methodologies in the ﬁeld are presentedin the recent books by N. Papamichael and N. Stylianopoulos [36] and by T. Driscolland L.N. Trefethen [45]. Various applications are described in [25, 28, 40, 47]. In the
past few years quadrilaterals and ring domains of increasing complexity have beenstudied by several authors [9, 8, 13, 39, 43].
We consider the problem of numerically determining the conformal modulus oncertain ring domains with elaborate boundary. Due to the structure of the boundary, the problem is numerically challenging. On the other hand, the ring domain ischaracterized by a triplet of parameters (
r,m,p
), its construction is recursive, andyet its conformal modulus can be explicitly given. Varying the parameter values orthe recursion level of the construction one can increase the computational challengeand therefore this family of domains forms a good set of test problems. In particular,error estimates can be given. For a ﬁgure of a domain in this family see Figure 3.1.The boundaries of these ring domains are point sets of dendrite type (i.e. continua
∗
Aalto University, Institute of Mathematics, P.O. Box 11100, FI00076 Aalto, FINLAND(
harri.hakula@aalto.fi
)
†
Aalto University, Institute of Mathematics, P.O. Box 11100, FI00076 Aalto, FINLAND(
antti.rasila@iki.fi
)
‡
Department of Mathematics and Statistics, FI20014 University of Turku, FINLAND(
vuorinen@utu.fi
)1
a r X i v : 1 5 0 1 . 0 6 7 6 5 v 1 [ m a t h . N A ] 2 7 J a n 2 0 1 5
2
H. HAKULA, A. RASILA AND M. VUORINEN
without loops) [48]. We apply here the
hp
FEM method developed in [22] for thecomputation of the moduli of these ring domains and report the accuracy of our algorithm. Furthermore, we use the algorithm of [21] to numerically approximate thecanonical conformal mapping of the above class of domains (see Figure 3.2 (c)). Theconjugate problem of the srcinal ring problem solved in the approximation of theconformal mapping can be interpreted as a simpliﬁed crack problem.Due to the pioneering work of I. Babuska and his coauthors [5, 6, 7] the opti
mal convergence rate of the
hp
FEM method is well studied and experimentally alsodemonstrated in some fundamental basic situations. Our main results are the computation the moduli of ring domains, given in the form of various error estimates, wherewe compare three error norms in the case of rings with elaborate boundary: (1) Exacterror (2) Auxiliary space estimate, based on
hp
space theory (3) So called reciprocalerror estimate, introduced in [22]. We also attain the nearly optimal convergence inaccordance with the theory of [5, 6, 7]. All these three error estimates behave in the
same way. Because this is the case, there is some justiﬁcation to use the estimates(2) and (3) also in the common case of applications when the exact value of the modulus is not know and hence estimate (1) is not available. We apply our methods tocompute the modulus of a quadrilateral considered by Bergweiler and Eremenko [10].The boundary of this domain has cusplike singularities.
1.1. Project background and history.
This paper is a culmination of a tenyear research project, arising from questions related to the work of Betsakos, Samuelsson and Vuorinen [11]. The srcinal goal of the project was to develop accurate numerical tools suitable for studying eﬀects of geometric transformations in functiontheory (see e.g. [17, 24]). Experimental work towards this goal was carried out by
Rasila and Vuorinen in the two small papers [37, 38], and further work along the same
lines was envisioned. However, it was quickly discovered that the AFEM package of Samuelsson used in the above papers is not optimal for studying very complex geometries arising from certain theoretical considerations, as the number of elementsused in such computations tends to become prohibitively large. This problem led usinto the higher order
hp
FEM algorithm implemented by Hakula. Eﬃciency of thismethod for numerical computation of conformal modulus had been established in thepapers [22] and [23], the latter of which deals with unbounded domains. Recently, an
implementation of this algorithms for the purpose of numerical conformal mappingwas presented in [21].
2. Preliminaries.
In this section central concepts to our discussion are introduced. The quantities of interest from function theory are related to numerical methods, and the error estimators arising from the basic principles are deﬁned.
2.1. Conformal Modulus.
A simplyconnected domain
D
in the complex plane
C
whose boundary is homeomorphic to the unit circle, is called a Jordan domain.A Jordan domain
D
, together with four distinct points
z
1
,z
2
,z
3
,z
4
in
∂D,
whichoccur in this order when traversing the boundary in the positive direction, is called a
quadrilateral
and denoted by (
D
1
;
z
1
,z
2
,z
3
,z
4
)
.
If
f
:
D
→
fD
is a conformal mappingonto a Jordan domain
fD
, then
f
has a homeomorphic extension to the closure
D
(also denoted by
f
). We say that the
conformal modulus
of (
D
;
z
1
,z
2
,z
3
,z
4
) is equalto
h >
0, if there exists a conformal mapping
f
of
D
onto the rectangle [0
,
1]
×
[0
,h
],with
f
(
z
1
) = 1 +
ih
,
f
(
z
2
) =
ih
,
f
(
z
3
) = 0 and
f
(
z
4
) = 1.It follows immediately from the deﬁnition that the conformal modulus is invariant
CONFORMAL MODULUS, SINGULARITIES AND CUSPS
3under conformal mappings, i.e.,
M
(
D
;
z
1
,z
2
,z
3
,z
4
) =
M
(
fD
;
f
(
z
1
)
,f
(
z
2
)
,f
(
z
3
)
,f
(
z
4
))
,
for any conformal mapping
f
:
D
→
f
(
D
) such that
D
and
f
(
D
) are Jordan domains.For a curve family Γ in the plane, we use the notation
M
(Γ) for its modulus[30]. For instance, if Γ is the family of all curves joining the opposite
b
sides withinthe rectangle [0
,a
]
×
[0
,b
]
,a,b >
0
,
then
M
(Γ) =
b/a.
If we consider the rectangleas a quadrilateral
Q
with distinguished points
a
+
ib,ib,
0
,a
we also have
M
(
Q
;
a
+
ib,ib,
0
,a
) =
b/a,
see [1, 30]. Given three sets
D,E,F
we use the notation ∆(
E,F
;
D
)for the family of all curves joining
E
with
F
in
D.
2.2. Modulus of a quadrilateral and Dirichlet integrals.
One can express the modulus of a quadrilateral (
D
;
z
1
,z
2
,z
3
,z
4
) in terms of the solution of theDirichletNeumann problem as follows. Let
γ
j
,
j
= 1
,
2
,
3
,
4 be the arcs of
∂D
between(
z
4
,z
1
)
,
(
z
1
,z
2
)
,
(
z
2
,z
3
)
,
(
z
3
,z
4
)
,
respectively. If
u
is the (unique) harmonic solutionof the DirichletNeumann problem with boundary values of
u
equal to 0 on
γ
2
, equalto 1 on
γ
4
and with
∂u/∂n
= 0 on
γ
1
∪
γ
3
,
then by [1, p. 65/Thm 4.5]:
M
(
D
;
z
1
,z
2
,z
3
,z
4
) =
D
∇
u

2
dxdy.
(2.1)The function
u
satisfying the above boundary conditions is called the
potential function
of the quadrilateral (
D
;
z
1
,z
2
,z
3
,z
4
).
2.3. Modulus of a ring domain and Dirichlet integrals.
Let
E
and
F
betwo disjoint compact sets in the extended complex plane
C
∞
. Then one of the sets
E, F
is bounded and without loss of generality we may assume that it is
E .
If both
E
and
F
are connected and the set
R
=
C
∞
\
(
E
∪
F
) is connected, then
R
is calleda
ring domain
. In this case
R
is a doubly connected plane domain. The
capacity
of
R
is deﬁned bycap
R
= inf
u
D
∇
u

2
dxdy,
where the inﬁmum is taken over all nonnegative, piecewise diﬀerentiable functions
u
with compact support in
R
∪
E
such that
u
= 1 on
E
. It is wellknown thatthere exists a unique harmonic function on
R
with boundary values 1 on
E
and 0on
F
. This function is called the potential function of the ring domain
R
, and itminimizes the above integral. In other words, the minimizer may be found by solvingthe Dirichlet problem for the Laplace equation in
R
with boundary values 1 on thebounded boundary component
E
and 0 on the other boundary component
F .
A ringdomain
R
can be mapped conformally onto the annulus
{
z
:
e
−
M
<

z

<
1
}
, where
M
=
M
(
R
) is the conformal modulus of the ring domain
R.
The modulus and capacityof a ring domain are connected by the simple identity
M
(
R
) = 2
π/
cap
R
. For moreinformation on the modulus of a ring domain and its applications in complex analysisthe reader is referred to [1, 25, 28, 36].
2.4. Hyperbolic Metrics.
The hyperbolic geometry in the unit disk is a powerful tool of classical complex analysis. We shall now brieﬂy review some of the mainfeatures of this geometry, necessary for what follows. First of all, the hyperbolicdistance between
x,y
∈
D
is given by
ρ
D
(
x,y
) = 2arsinh

x
−
y

(1
−
x

2
)(1
−
y

2
)
.
4
H. HAKULA, A. RASILA AND M. VUORINEN
In addition to the unit disk
D
, one usually also studies the upper half plane
H
as amodel of the hyperbolic geometry. For
x,y
∈
H
we have (
x
= (
x
1
,x
2
))
ρ
H
(
x,y
) = arcosh
1 +

x
−
y

2
2
x
2
y
2
.
If there is no danger of confusion, we denote both
ρ
H
(
z,w
) and
ρ
D
(
z,w
) simply by
ρ
(
z,w
). We assume that the reader is familiar with some basic facts about these geometries: geodesics, hyperbolic length minimizing curves, are circular arcs orthogonalto the boundary in each case.Let
z
1
,z
2
,z
3
,z
4
be distinct points in
C
. We deﬁne the
absolute (cross) ratio
by

z
1
,z
2
,z
3
,z
4

=

z
1
−
z
3

z
2
−
z
4

z
1
−
z
2

z
3
−
z
4

.
(2.2)This deﬁnition can be extended for
z
1
,z
2
,z
3
,z
4
∈
C
∞
by taking the limit. An important property of M¨obius transformations is that they preserve the absolute ratios,i.e.

f
(
z
1
)
,f
(
z
2
)
,f
(
z
3
)
,f
(
z
4
)

=

z
1
,z
2
,z
3
,z
4

,
if
f
:
C
∞
→
C
∞
is a M¨obius transformation. In fact, a mapping
f
:
C
∞
→
C
∞
is aM¨obius transformation if and only if
f
it is sensepreserving and preserves all absoluteratios.Both for (
D
,ρ
D
) and (
H
,ρ
H
) one can deﬁne the hyperbolic distance in terms of theabsolute ratio. Since the absolute ratio is invariant under M¨obius transformations,the hyperbolic metric also remains invariant under these transformations. In particular, any M¨obius transformation of
D
onto
H
preserves the hyperbolic distances. Astandard reference on hyperbolic metrics is [2].
2.5.
hp
FEM.
In this work the natural quantity of interest is always relatedto the Dirichlet energy. Of course, the ﬁnite element method (FEM) is an energyminimizing method and therefore an obvious choice. The continuous Galerkin
hp
FEM algorithm used throughout this paper is based on our earlier work [22]. Brief outline of the relevant features used in numerical examples below is: BabuˇskaSzabo type
p
elements, curved elements with blendingfunction mapping for exact geometry,rulebased meshing for geometrically graded meshes, and in the case of isotropic
p
distribution, hierarchical solution for all
p
. The main new feature considered here isthe introduction of auxiliary subspace techniques for error estimation.For the types of problems considered here, theoretically optimal conforming
hp
adaptivity is hard. The main diﬃculty lies in mesh adaptation since the desiredgeometric or exponential grading is not supported by standard data structures suchas Delaunay triangulations. Thus, the approach advocated here is a hybrid one, wherethe problem is ﬁrst solved using an
a priori
hp
algorithm after which the quality of the solution is estimated using error estimators speciﬁc both for the problem andthe method, provided the latter are available. For instance, the exact solution or forproblems concerning the conformal modulus the socalled reciprocal error estimator.The a priori algorithm is modiﬁed if the error indicators suggest modiﬁcations. If thisoccurs, the solution process is started anew.In the numerical examples below the computed results are measured with bothkinds of error estimators giving us high conﬁdence in the validity of the results andthe chosen methodology.
CONFORMAL MODULUS, SINGULARITIES AND CUSPS
5
2.5.1. Auxiliary Subspace Techniques.
Consider the abstract problem setting with
V
as the standard piecewise polynomial ﬁnite element space on some discretization
T
of the computational domain
D
. Assuming that the exact solution
u
∈
H
10
(
D
) has ﬁnite energy, we arrive at the approximation problem: Find ˆ
u
∈
V
such that
a
(ˆ
u,v
) =
l
(
v
) (=
a
(
u,v
))
,
∀
v
∈
V,
(2.3)where
a
(
·
,
·
) and
l
(
·
), are the bilinear form and the load potential, respectively. Additional degrees of freedom can be introduced by enriching the space
V
. This isaccomplished via introduction of an auxiliary subspace or “error space”
W
⊂
H
10
(
D
)such that
V
∩
W
=
{
0
}
. We can then deﬁne the error problem: Find
∈
V
such that
a
(
,v
) =
l
(
v
)
−
a
(ˆ
u,v
)(=
a
(
u
−
ˆ
u,v
))
,
∀
v
∈
W.
(2.4)In 2D the space
W
, that is, the additional unknowns, can be associated with elementedges and interiors. Thus, for
hp
methods this kind of error estimation is natural.For more details on optimal selection of auxiliary spaces, see [20].The solution
of (2.4) is called the
error function
. It has many useful propertiesfor both theoretical and practical considerations. In particular, the error function canbe numerically evaluated and analysed for any ﬁnite element solution. This propertywill be used in the following. By construction, the error function is identically zeroat the mesh points. In Figure 3.3 one instance of a contour plot of the error function(with a detail) is shown. This gives an excellent way to get a qualitative view of thesolution which can be used to reﬁne the discretization in the
hp
sense.Let us denote the error indicator by a pair (
e,b
), where
e
and
b
refer to addedpolynomial degrees on edges and element interiors, respectively. It is important tonotice that the estimator requires a solution of a linear system. Assuming that theenrichment is ﬁxed over the set of
p
problems, it is clear that the error indicatoris expensive for small values of
p
but becomes asymptotically less expensive as thevalue of
p
increases. Following the recommendation of [20], our choice in the sequelis (
e,b
) = (1
,
2) unless speciﬁed otherwise.
Remark 2.1.
In the case of
(0
,b
)
type or pure bubble indicators, the system is not connected and the elemental error indicators can be computed independently,and thus in parallel. Therefore in practical cases one is always interested in relative performance of
(0
,b
)
type indicators.
2.6. Reciprocal Identity and Error Estimation.
Let
Q
be a quadrilateraldeﬁned by points
z
1
,z
2
,z
3
,z
4
and boundary curves as in Section 2.1 above. Thefollowing reciprocal identity holds:
M
(
Q
;
z
1
,z
2
,z
3
,z
4
)
M
(
Q
;
z
2
,z
3
,z
4
,z
1
) = 1
.
(2.5)As in [22, 23], we shall use the test functional
M
(
Q
;
z
1
,z
2
,z
3
,z
4
)
M
(
Q
;
z
2
,z
3
,z
4
,z
1
)
−
1
(2.6)which by (2.5) vanishes identically, as an error estimate.As noted above, the error function
can be analysed in the sense of FEMsolutions. Our goal is to relate the error function given by auxiliary space techniquesand the reciprocal identity arising naturally from the geometry of the problem. Letus ﬁrst deﬁne the energy of the error function
as
E
(
) =
D
∇

2
dxdy.
(2.7)